Along comes a Chrome Book which cheerfully connects wirelessly to The Internet, but under "Google Cloud Print" -> "Manage" the Chrome Book cannot see any printers.
If I connect the Chrome Book via a USB cable direct to the laser printer it can see it.

My question is:
is there any way I can get the Chrome Book to print 'wirelessly' via the Windows 7 PC to the laser printer?

Sorry about that. I don't see why a wireless connection should be needed as long as you're on the same network. I did need a third party utility on the PC with a shared printer to be able to print from my IPad before I got a wireless printer. That's what I thought using Chrome on the PC with a Google account would provide as described in the link.

Does your Brother printer have an ethernet port? In other words, can you connect it with an ethernet cable to your router? If so, you could make it a network printer, and access it that way from your Windows 7 computer and from your Chromebook.

By doing it this way, you would eliminate the need for the Chromebook to negotiate its way through Window 7 security. My guess is that this is where the process is failing.

I don't see any difference between a chrome book vs windows laptop connecting to a printer on a network. All you should have to do is make sure the chrome book is on the network, and through the network on the chrome book just add the printer driver...like you would any other laptop to print from the network. I have 3 laptops setup this way and all print via the wifi of the network.

My printer is an old HP 5610v with no built in wifi or ethernet connection. I've had this setup for years and years....works great.

I don't see any difference between a chrome book vs windows laptop connecting to a printer on a network. All you should have to do is make sure the chrome book is on the network, and through the network on the chrome book just add the printer driver...like you would any other laptop to print from the network. I have 3 laptops setup this way and all print via the wifi of the network.

My printer is an old HP 5610v with no built in wifi or ethernet connection. I've had this setup for years and years....works great.

How can I convince the Chromebook that there is really a printer available?

I have found this thread (see the top post) which seems to indicate that the problem is non-trivial... I have to get a USB wireless adapter to test it with.

Could it be simpler to just pull the document down from the cloud on your windows machine and then print? Maybe not as expedient as printing directly, but it would work. This is how we "network" the printer at my house: my daughter creates her document for school, saves it to a USB drive, then "sneakernet's" it to my desktop which has a laser printer attached and, voila, printing ensues.
Just a thought.